1. Field of the Invention
The present invention relates to an application of chaos theory to controlling in synchronization at least two systems and, more particularly, relates to a pseudoperiodic drive for controlling in synchronization at least two systems.
2. Description of the Related Art
FIG. 1 illustrates a conventional control system for controlling two response systems, a first system 110 and second system 120, by a conventional controller 130. The conventional controller 130 determines the location of the state variables of the first and second systems 110 and 120 from feedback therefrom. With knowledge of the location of the state variables of each response system, the conventional controller 130 controls the response systems to place the systems in synchronization with one another. By synchronization, it is meant that both systems are doing the same thing at the same time.
However, conventional controllers have drawbacks. First, for example, the conventional controller 130 is often limited to controlling only linear response systems 110 and 120. Second, in order to synchronize two systems, the conventional controller 130 requires the complexity of separate outputs to each system and a feedback input from each system. Third, an understanding of the response characteristics of the response systems is often built into the mechanism of the conventional system controller 130. Thus, the conventional controller 130 is often unable to effectively control the systems when a precise understanding of the systems is unobtainable. Furthermore, if one of the feedback or control paths between the controller and the response systems is disrupted due to an intermittent failure, the response system could rapidly diverge from its desired path. This is because the controller depends heavily on the understanding of the response systems. Furthermore, in a worst case scenario, the control systems could become irreversibly unstable.
Most man-made mechanical and electronic systems rely for their operation on linear behavior or simple nonlinear behavior, such as fixed points (stationary states) or periodic (cyclic) variations. Primarily this comes from an ignorance of more complex behavior and, subsequently, a lack of control over such behavior. Recently, in many areas of science, mathematics and engineering a new kind of behavior has become recognized as being generic in most nonlinear systems. This is called chaos. Being generic means that given any nonlinear dynamical system, it is very likely that in some regime of its parameter settings it has motion of this sort.
Systems evolving chaotically display a sensitivity to initial conditions. That is, two nearly identical systems started at slightly different values of their state variables will soon evolve to values which are vastly different and the systems will become completely uncorrelated, even though the overall patterns of behavior will remain the same. This makes the systems nonperiodic (no cycles whatsoever), a quality which can be exploited.
There are also other types of behavior in nonlinear systems which can lead to systems being out of phase or out of synchronization with each other. This behavior is multiple-period behavior and can occur in systems that are driven with a periodic signal or force. The out-of-phase problem has not been resolved with existing techniques. Several terms and concepts will now be defined which are necessary in understanding the control of nonlinear systems.
FIGS. 2(a) and 2(b) illustrate a moving pendulum and a phase plot of the motion of the moving pendulum to explain the principles of phase plots. FIG. 2(b) illustrates a phase plot of the motion of the pendulum illustrated in FIG. 2(a). The phase plot illustrated in FIG. 2(b) is a two-dimensional (x-y) plot, where the x-axis corresponds to one variable of interest (position of the pendulum) and the y-axis corresponds to another variable of interest (velocity of the pendulum) among a plurality of variables which describe the motion of the pendulum. Multi-dimensional phase space plots of a response system are possible if more than two dynamical variables are necessary to describe the system. Phase space is often referred to as state space.
A point in state or phase space describes, at once, the values of the dynamical variables of the system. Correspondingly, the path the point takes in phase space as the system evolves in time traces the behavior of the system. This path is often referred to as a trajectory. The trajectory describes the history of the system.
Phase space plots can be generated on the screen of a computer using equations to describe the motion of a system. It is also possible that an oscilloscope can be used to display a phase plot for two measured electrical quantities from a circuit (which is also a dynamical system) along x and y axes of the oscilloscope screen. On the screen, the changes of the x and y variables will create a pattern which will be a projection of the phase space trajectory onto two dimensions. This plot will be in real time because the oscilloscope time sweep is not used in an x/y plot of the voltage. It should be noted that a real time plot can be obtained by plotting a voltage (x(t)) vs. its delayed value (x(t+.delta.t)), its derivative x(t), or its integral x(t)dt.
Many physical systems often have dissipation, energy losses, or friction occurring naturally during their evolution. This manifests itself by having the phase space trajectory settle into motion on a particular path in the state space. In the damped linear pendulum, which is driven by a periodic force (e.g. someone being pushed gently on a swing), this results in the motion settling into a simple closed loop as shown in FIG. 2. For the linear pendulum, starting the system at any point in phase space will still result in its ending up on the same trajectory. In fact, any two pendula started at different initial points (called initial conditions) will end up having their phase space points not only on the same trajectory but, eventually, at the same moving point on the trajectory. In other words, they will have identical motion, and the same moving phase space point will describe both pendula. The final trajectory is called an attractor.
The above scenario is specific in some ways to linear systems, but nonlinear systems have analogous phase space properties. Nonlinear dynamical systems with dissipation will also have their final motion lie on some path (trajectory) confined to a much smaller region of phase space than is accessible to the system as a whole. However, in this case not all the initial conditions will lead to the same attractor. Nonlinear systems can have several different attractors present at the same time. Which one the system will converge to will depend on where it started.
For example, in the case where a system consists of a ball under the influence of gravity and rolling friction riding on a surface that has multiple valleys, which valley the ball ends up in will depend on where it started and with what velocity it started (i.e., its phase space initial condition). In this example the final attractor is simply a fixed point (the ball comes to rest). The set of all phase initial points that go to a particular fixed point attractor is called the basin of attraction for that attractor.
These concepts generalize from this example to attractors that are not fixed points, e.g., loops, chaotic attractors, etc. Several can be present at the same time and each will have a basin of attraction. FIG. 3 shows schematically a phase space with two attractors. FIG. 4(a) through 4(d) respectively show representations of various attractors that can be present: fixed point, periodic (trajectories that close on themselves), quasiperiodic (trajectories with two or more incommensurate periodic frequencies), and chaotic (trajectories with no periodicities and often fractal structure). However simple or complex an attractor, once the system's motion converges to it, the motion remains on it, unless perturbed by an outside force (that is one not associated with the dynamics of the system and its attractors and not in force during the system's convergence to an attractor). If the outside force is small and the system remains in the same basin, then when the outside force is turned off the motion will again settle down to the same attractor. If the system's phase space point is moved to another basin it will settle onto another attractor.
An important fact to note is that the boundaries delimiting adjacent basins of attraction must in some way be associated with an unstable motion. This is because two points, however close but on opposite sides of the boundary must eventually go to different attractors far away from each other.
The entire phase space is divided into basins of attraction, one for each attractor. However, unlike the linear pendulum, there are attractors on which two phase space points (representing two identical systems started at different initial conditions) may never move about the attractor and merge to one moving point. A chaotic attractor is of this nature. In fact, at any time there is a separate point moving on the attractor corresponding to each initial condition in the basin of attraction. There are also periodic attractors which have two or more phase space points moving on them corresponding to different entire sets of initial conditions in the basin. These latter systems will now be discussed.
Many nonlinear systems when driven with periodic signals or forces have regimes of their parameters where they behave periodically, but with a period that is some multiple of the period of the driving signal. This has been known for decades and is behavior that is generic to nonlinear systems. What this means is that the systems take n driving periods (where n is some integer) to return to their starting points. For example, when n=2 the systems will have a period of repetition that is twice that of the driving signal. This latter case is often referred to as a period-doubled system. Many other period multiplicities are possible. The driven system can be at any of several different points moving on the attractor during each cycle of the drive. There are n such points for a period-n behavior. Which one the system is at will depend on where it started when the drive signal was turned on. All the points in the same basin of attraction end up on the same attractor, but the basin itself can be subdivided into domains. Points started in the same domain will end up converging to the same point moving around the attractor. Points started in different domains will end up on different points moving around the same attractor. Like adjacent basins, adjacent domains have a common boundary which is associated with an unstable motion.
The salient point is that if several nonlinear systems are being driven by a periodic signal, they can behave in a multiple-period fashion and be permanently out of phase with each other even though they are on the same attractor and undergoing the same type of motion. This is shown in FIG. 5 for a period-doubled motion. This out-of-phase situation is stable. Small perturbations (outside forces, as above) will not change it.
There are instances when out-of-phase behavior is undesirable. However, there may be simultaneous instances when multiple-period behavior is required (perhaps because of the shape of the waveform needed) or unavoidable. For example, this may occur in robotics, laser arrays, or physiology. In robotics, if nonlinear materials or actuators are being used, and these need to be driven by a timing signal, then any multiple-period behaving parts could get out of phase. In relation to lasers, if a set of lasers is being operated in a period-doubled regime to get a half-frequency component, then these should be in phase to maximize output intensity. In physiology, many tissues and organs are nonlinear. In order to keep such systems in synchronization with each other in many driven circumstances, the above in-phase requirements may be necessary.
Furthermore, it is well known that systems with several final forms of behavior (attractors) can have fractal basin boundaries. That is, the boundaries are complexly intertwined, in a fractal fashion. In this case, prediction of the final system state can be very difficult, since the fractal structure gives an uncertainty to determining the domain of the initial conditions. This uncertainty is difficult to eliminate. Decreasing the number of final attractors, especially those differing by phase shifts in the system's motion, will diminish or completely eliminate this problem.